We note a crucial difference between previous studies relating long-range interactions, such as a dipole-dipole interaction, and the formation of vortices in the presence of an artificial magnetic field [1]}, [2]}. In these earlier works, the dipole-dipole interaction, an artificial magnetic field, and vortices all appear in the same (physical) two-dimensional plane. Our result, on the other hand, considers a long-range (intertube) interaction in the \(x\) -\(y\)  plane, but the vortices appear in the reciprocal \(x\) -\(p_y\)  plane where the artificial magnetic field is realized.
